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Noisy-Or Nodes To avoid specifying the complete CPT, special nodes that make assumptions about the style of interaction can be used. A noisy­or node assumes that the parents are independent causes that are noisy, i.e. there is some probability that they will not cause the effect. The noise parameter for each cause indicates the probability it will not cause the effect. Probability that the effect is not present is the product of the noise parameters of all the parent nodes that are true (since independence is assumed). P(Fever|Cold) =0.4,P(Fever|Flu) =0.8,P(Fever| Malaria)=0.9 P(Fever | Cold  Flu  ¬Malaria) = 1-0.6 * 0.2 = 0.88 Number of parameters needed is linear in fan­in rather than exponential.

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Independencies If removing a subset of nodes S from the network renders nodes X i and X j disconnected, then X i and X j are independent given S, i.e. P(X i | X j, S) = P(X i | S) However, this is too strict a criteria for conditional independence since two nodes will still be considered independent if there simply exists some variable that depends on both. (i.e. Burglary and Earthquake should be considered independent since the both cause Alarm)

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Unless we know something about a common effect of two “independent causes” or a descendent of a common effect, then they can be considered independent. For example,if we know nothing else, Earthquake and Burglary are independent. However, if we have information about a common effect (or descendent thereof) then the two “independent” causes become probabilistically linked since evidence for one cause can “explain away” the other. If we know the alarm went off, then it makes earthquake and burglary dependent since evidence for earthquake decreases belief in burglary and vice versa.

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Types of Connections Given a triplet of variables x, y, z where x is connected to z via y, there are 3 possible connection types: –tail­to­tail: x  y  z –head­to­tail: x  y  z, or x  y  z –head­to­head: x  y  z For tail­to­tail and head­to­tail connections, x and z are independent given y. For head­to­head connections, x and z are “marginally independent” but may become dependent given the value of y or one of its descendents (through “explaining away”).

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Separation A subset of variables S is said to separate X from Y if all (undirected) paths between X and Y are separated by S. A path P is separated by a subset of variables S if at least one pair of successive links along P is blocked by S. Two links meeting head­to­tail or tail­to­tail at a node Z are blocked by S if Z is in S. Two links meeting head­to­head at a node Z are blocked by S if neither Z nor any of its descendants are in S.

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Probabilistic Inference Given known values for some evidence variables, we want to determine the posterior probability of of some query variables. Example: Given that John calls, what is the probability that there is a Burglary? John calls 90% of the time there is a burglary and the alarm detects 94% of burglaries, so people generally think it should be fairly high (80­90%). But this ignores the prior probability of John calling. John also calls 5% of the time when there is no alarm. So over the course of 1,000 days we expect one burglary and John will probably call. But John will also call with a false report 50 times during 1,000 days on average. So the call is about 50 times more likely to be a false report P(Burglary | JohnCalls) ~ 0.02. Actual probability is 0.016 since the alarm is not perfect (an earthquake could have set it off or it could have just went off on its own). Of course even if there was no alarm and John called incorrectly, there could have been an undetected burglary anyway, but this is very unlikely.

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Inference Algorithms Most inference algorithms for Bayes nets are not goal­directed and calculate posterior probabilities for all other variables. In general, the problem of Bayes net inference is NP­hard (exponential in the size of the graph).

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Polytree Inference For singly­connected networks or polytrees, in which there are no undirected loops (there is at most one undirected path between any two nodes), polynomial (linear) time algorithms are known. Details of inference algorithms are somewhat mathematically complex, but algorithms for polytrees are structurally quite simple and employ simple propagation of values through the graph.

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Belief Propagation Belief propogation and updating involves transmitting two types of messages between neighboring nodes: – messages are sent from children to parents and involve the strength of evidential support for a node. –  messages are sent from parents to children and involve the strength of causal support.

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Propagation Details Each node B acts as a simple processor which maintains a vector (B) for the total evidential support for each value of the corresponding variable and an analagous vector  (B) for the total causal support. The belief vector BEL(B) for a node, which maintains the probability for each value, is calculated as the normalized product: BEL(B) =  (B)  (B)

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Propogation Details (cont.) Computation at each node involve and  message vectors sent between nodes and consists of simple matrix calculations using the CPT to update belief (the and  node vectors) for each node based on new evidence. Assumes CPT for each node is a matrix (M) with a column for each value of the variable and a row for each conditioning case (all rows must sum to 1).